For \(c\ge 3\) , \(T_c(n)=O(n^{1/2}\operatorname{poly}{(c)})\)  has been proved [1]}, and therefore for \(c=O(n^{o(1)})\)  the \(c\) -edge-connected components can be maintained under deletions in \(O(m)+\widetilde{O}(n^{3/2+o(1)})\)  total time.
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